Solving simultaneous equations by elimination

Simultaneous equations are when we have two or more equations with multiple unknowns (letters that represent numbers we don’t know yet). The goal is to find the values of these letters (variables) that make all the equations true at the same time.

When can we solve using elimination?

We can use the elimination method when the equations are both linear (in the form ax + by = c), and we have the same number of equations as unknowns ( for example, two equations with two unknowns, x and y).

Basic steps to solve

The key steps to solve a simultaneous equation with two unknowns by elimination are:

Find both equations in the form ax + by = c.
Multiply one or both equations by a number so that the coefficients (the numbers in front of the variables) of one of the variables are the same (or opposites). This allows us to cancel out that variable.
Add or subtract one equation from another. This involves adding or subtracting each term individually to form a new equation with just one variable.
Solve this new equation to find the value of one variable.
Substitute this value back into one of the original equations to find the value of the other variable.

Like with most concepts, it’s much easier to see with some examples.

Examples

Example: solve the simultaneous equations 2x + 3y = 16 and 4x - y = 2.

First, we have the two equations:
- 2x + 3y = 16 (Equation 1)
- 4x - y = 2 (Equation 2)
We want to eliminate one of the variables. You can do whichever you want, but I will eliminate y here. To do this, we can multiply Equation 2 by 3 so that the coefficient of y in both equations will be opposites:
- 2x + 3y = 16 (Equation 1)
- 12x - 3y = 6 (Equation 2 multiplied by 3)
Add the two equations together to eliminate the y variable:
- (2x + 12x) + (3y - 3y) = 16 + 6
- 14x + 0 = 22
- 14x = 22
Solve for x:
- x = \frac{22}{14} = \frac{11}{7}
substitute x = \frac{11}{7} back into one of the original equations to find y. You can use whichever equation you like: I’ll use Equation 1:
- 2(\frac{11}{7}) + 3y = 16
- \frac{22}{7} + 3y = 16
- 3y = 16 - \frac{22}{7}
- 3y = \frac{112}{7} - \frac{22}{7}
- 3y = \frac{90}{7}
- y = \frac{90}{21} = \frac{30}{7}
So the solution to the simultaneous equations is:
- x = \frac{11}{7}
- y = \frac{30}{7}

Checking your solution

To check if your solution is correct, substitute the values of x and y back into one or both of the original equations to see if they work!

flashcards

Question	Answer
What are simultaneous equations?	Equations with two or more unknowns where the goal is to find values that make all equations true at the same time.
When can we use the elimination method to solve simultaneous equations?	When both equations are linear (in the form ax + by = c) and there is the same number of equations as unknowns.
What is the first step in solving by elimination?	Rewrite both equations in the form ax + by = c.
After arranging equations in the form ax + by = c, what do you do next?	Multiply one or both equations so that the coefficients of one variable are the same or opposites.
After making coefficients the same or opposites, what operation is performed?	Add or subtract the equations term-by-term to eliminate one variable, creating a new equation with one variable.
After you have a new equation with one variable, what do you do?	Solve it to find the value of that variable.
After finding one variable’s value, how do you find the other variable?	Substitute that value back into one of the original equations and solve for the other variable.
In Example: 2x + 3y = 16 and 4x - y = 2, which step eliminates y?	Multiply equation 2 by 3 to get 12x - 3y = 6, so the y coefficients are opposites (+3y and -3y).
In the example 2x + 3y = 16 and 12x - 3y = 6, what happens when you add them?	14x + 0 = 22, which gives 14x = 22.
What is the x solution in the example 2x + 3y = 16 and 4x - y = 2?	x = \frac{11}{7}
After finding x = \frac{11}{7}, how do you find y in the example?	Substitute into equation 1: 2(\frac{11}{7}) + 3y = 16, solving gives y = \frac{30}{7}.
How do you check if a solution to simultaneous equations is correct?	Substitute the values of x and y back into one or both original equations to see if they work.